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2/(16+6*7^(1/2))^(1/2)

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2/(16+6*7^(1/2))^(1/2)  [#permalink]

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New post 12 May 2017, 04:44
1
5
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A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

57% (01:47) correct 43% (02:13) wrong based on 194 sessions

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2/(16+6*7^(1/2))^(1/2)  [#permalink]

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New post 12 May 2017, 06:58
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IF x = \(\frac{2}{\sqrt{16+6\sqrt{7}}}\)
Squaring

x^2 = \(\frac{4}{{16+6\sqrt{7}}}\)

Multiplying and dividing by conjugate surd
= \(\frac{{16-6\sqrt{7}}}{16^2 - 36*7}\) (256 - 252 = 4)

= \(\frac{{16-6\sqrt{7}}}4\)

= \(16-6\sqrt{7}\)

Hence, we need the value of x, we get the value as \(\sqrt{16-6\sqrt{7}}\) (Option D)
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Re: 2/(16+6*7^(1/2))^(1/2)  [#permalink]

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New post 01 Jun 2017, 00:15
Bunuel wrote:
\(\frac{2}{\sqrt{16+6\sqrt{7}}}\)


A. 1/27

B. \(\frac{4}{16+6\sqrt{7}}\)

C. \(\frac{\sqrt{16-6\sqrt{7}}}{4}\)

D. \(\sqrt{16-6\sqrt{7}}\)

E. \(\sqrt{16+6\sqrt{7}}\)


= \(\frac{2}{\sqrt{16+6\sqrt{7}}}\)

= \(\frac{2}{\sqrt{9+7+2.3\sqrt{7}}}\)

= \(\frac{2}{\sqrt{3^2+\sqrt{7}^2+2.3\sqrt{7}}}\)

Quote:
\(a^2+b^2+2ab=(a+b)^2\)


= \(\frac{2}{\sqrt{(3+\sqrt{7})^2}}\)

= \(\frac{2}{{3+\sqrt{7}}}\)

Quote:
Rationalize


= \(\frac{2}{{3+\sqrt{7}}}*\frac{3-\sqrt{7}}{{3-\sqrt{7}}}\)

= \(3-\sqrt{7}\)

= \(\sqrt{16-6\sqrt{7}}\)

D.
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Re: 2/(16+6*7^(1/2))^(1/2)  [#permalink]

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New post 04 Jun 2017, 19:53
pushputkc,

can you pls elaborate what you did after squaring both sides (i.e. x^2)?
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Re: 2/(16+6*7^(1/2))^(1/2)  [#permalink]

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New post 04 Jun 2017, 23:07
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mrdlee23 wrote:
pushputkc,

can you pls elaborate what you did after squaring both sides (i.e. x^2)?


Hi, let me try.

First of all, whats a 'Conjugate'? A number of the form a + √b has its conjugate as a - √b and vice versa. Conjugates are useful in certain cases as in this question.

After writing x^2 = 4/(16+6√7) .. what he has done is multiplied both numerator and denominator by the conjugate of 16+6√7, i.e., multiplied both numerator and denominator by 16-6√7.. this will help in eliminating the √ sign from the denominator, thus denominator will be 'rationalised' (rational number) and we are likely to get an answer in the form as given in the question options.
So here is what he did:

x^2 = 4/(16+6√7)

Multiply both numerator and denominator by 16-6√7..

x^2 = 4*(16-6√7)/(16+6√7)*(16-6√7)
= 4 (16-6√7)/[(16)^2- (6√7)^2] (denominator becomes (a+b)(a-b) = a^2 - b^2)
= 4 (16-6√7)/(256-252)
=4 (16-6√7)/4 = 16-6√7

So x^2 simplifies to 16-6√7
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Re: 2/(16+6*7^(1/2))^(1/2)  [#permalink]

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Re: 2/(16+6*7^(1/2))^(1/2)   [#permalink] 10 Jan 2019, 08:17
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